Integral Transform and Segal-Bargmann Representation Associated to q-Charlier Polynomials

Let $\mu_p^{(q)}$ be the q-deformed Poisson measure in the sense of Saitoh Yoshida and $\nu_p$ be the measure given by Equation \eqref{eq:nu-q}. In this short paper, we introduce the q-deformed analogue of the Segal-Bargmann transform associated with $\mu_p^{(q)}$. We prove that our Segal-Bargmann transform is a unitary map of $L^2(\mu_p^{(q)})$ onto the q-deformed Hardy space ${\cal H}^2(\nu_q)$. Moreover, we give the Segal-Bargmann representation of the multiplication operator by $x$ in $L^2(\mu_p^{(q)})$, which is a linear combination of the q-creation, q-annihilation, q-number, and scalar operators.